Sure, let's solve the problems one by one.

Problem 6:

We have an arithmetic sequence with $a_1 = 25$ and $a_6 = 13$. We want to find the positive terms.

The general formula for the $n$-th term in an arithmetic sequence is: $a_n = a_1 + (n-1) \cdot d$ where $d$ is the common difference.

Given: $a_6 = 13 = a_1 + 5d$ $13 = 25 + 5d$ $5d = 13 - 25 = -12$ $d = -\frac{12}{5} = -2.4$

To find the positive terms, we need: $a_n = 25 + (n-1)(-2.4) > 0$ $25 - 2.4(n-1) > 0$ $25 > 2.4(n-1)$ $\frac{25}{2.4} > n-1$ $n-1 < \frac{125}{12}$ $n < \frac{125}{12} + 1 \approx 10.42$

Given options, there are 9 positive terms.

Answer: A) 9

Problem 7:

We need the function graph to pass through the origin $(0, 0)$.

Check each option:

• $y = x^2 + 1$ gives $y(0) = 1$, not passing through the origin.
• $y = x^2 + 4x$ gives $y(0) = 0+0 = 0$, passes through the origin.
• $y = x^2 - 2x + 1$ gives $y(0) = 1$, not passing through the origin.
• $y = 2 - x^2$ gives $y(0) = 2$, not passing through the origin.

Answer: Б) $y = x^2 + 4x$

Problem 8:

For the equation $2x^2 + 4x - 7 = 0$, using Vieta's formulas, $x_1 + x_2 = -\frac{b}{a} = -2$ and $x_1x_2 = \frac{c}{a} = -\frac{7}{2}$.

Calculate $x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1x_2$: $x_1^2 + x_2^2 = (-2)^2 - 2(-\frac{7}{2})$ $= 4 + 7$ $= 11$

Calculate $x_1^2 + x_2^2 - 4x_1x_2$: $= 11 + 14 = 25$

Answer: Г) 25

Problem 9:

Using the intersecting chords theorem: $AP \cdot PB = CP \cdot PD$ $6 \cdot 4 = 2 \cdot PD$ $24 = 2 \cdot PD$ $PD = 12$

Answer: В) 12

Problem 10:

The area of the rhombus given diagonals $d_1 = 6$ and $d_2 = 8$ is: $A = \frac{d_1 \cdot d_2}{2} = \frac{6 \cdot 8}{2} = 24 \, \text{cm}^2$

The area of the quadrilateral whose vertices are the midpoints of the rhombus is half the area of the rhombus.

Answer: Б) 24